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Boundary Layer Estimates in Stochastic Homogenization

Peter Bella, Julian Fischer, Marc Josien, Claudia Raithel

Abstract

We prove quantitative decay estimates for the boundary layer corrector in stochastic homogenization in the case of a half-space boundary. Our estimates are of optimal order and show that the gradient of the boundary layer corrector features nearly fluctuation-order decay; its expected value decays even one order faster. As a corollary, we deduce estimates on the accuracy of the representative volume element method for the computation of effective coefficients: our understanding of the decay of boundary layers enables us to improve the order of convergence of the RVE method for $d\geq 3$.

Boundary Layer Estimates in Stochastic Homogenization

Abstract

We prove quantitative decay estimates for the boundary layer corrector in stochastic homogenization in the case of a half-space boundary. Our estimates are of optimal order and show that the gradient of the boundary layer corrector features nearly fluctuation-order decay; its expected value decays even one order faster. As a corollary, we deduce estimates on the accuracy of the representative volume element method for the computation of effective coefficients: our understanding of the decay of boundary layers enables us to improve the order of convergence of the RVE method for .
Paper Structure (32 sections, 31 theorems, 269 equations)

This paper contains 32 sections, 31 theorems, 269 equations.

Table of Contents

  1. Introduction
  2. Discussion of main results and strategy
  3. Setting of our results
  4. Assumptions on the ensemble.
  5. Theorem \ref{['thm_1']}: Optimal decay estimate for the boundary layer
  6. Theorem \ref{['thm_2']}: Improved error estimate for representative volume element method.
  7. Overview of our arguments
  8. Strategy for Theorem \ref{['thm_1']}
  9. Step 1: A suboptimal quenched decay for $|\theta_i(x)|$ away from $\partial \mathbb{H}^d_+$.
  10. Step 2: Iterative improvement of the decay estimate.
  11. Argument for Proposition \ref{['iterative_lemma']}: Iterative improvement of decay estimate for the boundary layer
  12. Proof of Theorem \ref{['thm_2']}
  13. Argument for Proposition \ref{['intermediate_lemma']}: A suboptimal decay estimate for the boundary layer
  14. Proof of Proposition \ref{['intermediate_lemma']}
  15. Proof of auxiliary lemmas for Proposition \ref{['intermediate_lemma']}
  16. ...and 17 more sections

Key Result

Theorem 1

Let $d\geq 3$ and $\langle\cdot\rangle$ be an ensemble of coefficient fields on $\mathbb{R}^d$ that satisfies the assumptions (A1)-(A4). Denote by $\theta_i \in H^1_{\rm{loc}}(\mathbb{H}^d_+ ; \mathbb{R}^d)$ the half-space corrector, that is, the uniqueUniqueness follows from the Liouville principle Then there exists a random field $\mathcal{C}(a,x)$, defined on $\Omega \times \mathbb{H}^d_+$, tha

Theorems & Definitions (55)

  • Theorem 1
  • Theorem 2
  • Proposition 1
  • Proposition 2
  • Corollary 3
  • Lemma 4
  • proof
  • Proposition 5
  • Proposition 6
  • proof : Proof of Proposition \ref{['iterative_lemma']}
  • ...and 45 more